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The velocity component at any t along horizontal ( x-axis) is v x = u x + a x t The resultant of these two components gives the velocity of the projectile at that instant t, as shown in Figure 2.41. (3) Resultant Velocity (Velocity of projectile at any time): At any instant t, the projectile has velocity components along both x-axis and y-axis. The above equation implies that the range R is directly proportional to the initial velocity u and inversely proportional to acceleration due to gravity g. Here, s x = R (range), u x = u, a = 0 (no horizontal acceleration) T is time of flight. (2) Horizontal range: The horizontal distance covered by the projectile from the foot of the tower to the point where the projectile hits the ground is called horizontal range. If one ball falls vertically and another ball is projected horizontally with some velocity, both the balls will reach the bottom at the same time. Thus, the time of flight for projectile motion depends on the height of the tower, but is independent of the horizontal velocity of projection. Here s y = h, t = T, u y = 0 (i.e., no initial vertical velocity).
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We know that s y = u yt + 1/2 at 2 for vertical motion. Let T be the time taken by the projectile to hit the ground, after being thrown horizontally from the tower.
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(1) Time of Flight: The time taken for the projectile to complete its trajectory or time taken by the projectile to hit the ground is called time of flight.Ĭonsider the example of a tower and projectile. Thus, the path followed by the projectile is a parabola (curve OPA in the Figure 2.39). Substituting the value of t from equation (2.23) in equation (2.24) we haveĮquation (2.25) is the equation of a parabola. Here u y = 0 (initial velocity has no downward component), a = g (we choose the +ve y-axis in downward direction), and distance y at time t The distance traveled by the projectile at a time t is given by the equation x = u xt + 1/2 at 2. So, the initial velocity u x remains constant throughout the motion The particle has zero acceleration along x direction. Since this is two-dimensional motion, the velocity will have both horizontal component u x and vertical component u y. We can apply the kinematic equations along the x direction and y direction separately. Let the ball take time t to reach the ground at point A, Then the horizontal distance travelled by the ball is x (t) = x, and the vertical distance travelled is y (t) = y Thus, under the combined effect the ball moves along the path OPA. The horizontal range's unit is meter (m).Consider a projectile, say a ball, thrown horizontally with an initial velocity from the top of a tower of height h (Figure 2.39).Īs the ball moves, it covers a horizontal distance due to its uniform horizontal velocity u, and a vertical downward distance because of constant acceleration due to gravity g. After that, the horizontal range depends on the initial velocity, the projected angle θ, and the acceleration happening due to gravity. Moreover, it would move before it approaches the same vertical position as it began. A projectile's horizontal range is the length along with the horizontal level. This depends on the angle of projection and the initial velocity of the projectile.Ī projectile is an object that provides an initial velocity, and gravity works on it. The time it needs from an object to be thrown and land is named the time of flight. Objects projected from and land on a similar horizontal surface will have a vertically balanced path. Projectile motion is when an object passes in the parabolic path its path is termed its trajectory.
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The horizontal displacement of the object projected is called the projectile range and depends on the object's initial velocity. Hint: When the projectile touches a vertical velocity of zero, range is the highest height of the projectile, and then gravity will get over and accelerate the object descending.